3.199 \(\int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac{3 x}{-x^2+e^{2 i a}}+\frac{e^{2 i a}}{x \left (-x^2+e^{2 i a}\right )}-2 e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right ) \]

[Out]

E^((2*I)*a)/(x*(E^((2*I)*a) - x^2)) - (3*x)/(E^((2*I)*a) - x^2) - (2*ArcTanh[x/E^(I*a)])/E^(I*a)

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Rubi [F]  time = 0.0478845, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]^2/x^2,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]^2/x^2, x]

Rubi steps

\begin{align*} \int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx &=\int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.121487, size = 72, normalized size = 1.12 \[ \frac{2 x (\cos (a)-i \sin (a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-2 \cos (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+2 i \sin (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]^2/x^2,x]

[Out]

x^(-1) - 2*ArcTanh[x*(Cos[a] - I*Sin[a])]*Cos[a] + (2*I)*ArcTanh[x*(Cos[a] - I*Sin[a])]*Sin[a] + (2*x*(Cos[a]
- I*Sin[a]))/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])

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Maple [A]  time = 0.069, size = 62, normalized size = 1. \begin{align*}{x}^{-1}-2\,{\frac{1}{x \left ( \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1 \right ) }}-{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{{{\rm e}^{ia}}}}+{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{{{\rm e}^{ia}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x))^2/x^2,x)

[Out]

1/x-2/x/(exp(I*(a+I*ln(x)))^2-1)-1/exp(I*a)*ln(exp(I*a)+x)+1/exp(I*a)*ln(exp(I*a)-x)

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Maxima [B]  time = 1.15388, size = 385, normalized size = 6.02 \begin{align*} -\frac{2 \,{\left ({\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{3} +{\left ({\left (2 \,{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (2 \,{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x - 6 \, x^{2} +{\left (x^{3}{\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} -{\left ({\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) -{\left (x^{3}{\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} -{\left ({\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )}{2 \, x^{3} - x{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="maxima")

[Out]

-(2*((I*cos(a) + sin(a))*arctan2(sin(a), x + cos(a)) + (I*cos(a) + sin(a))*arctan2(sin(a), x - cos(a)))*x^3 +
((2*(-I*cos(a) - sin(a))*cos(2*a) + (2*cos(a) - 2*I*sin(a))*sin(2*a))*arctan2(sin(a), x + cos(a)) + (2*(-I*cos
(a) - sin(a))*cos(2*a) + (2*cos(a) - 2*I*sin(a))*sin(2*a))*arctan2(sin(a), x - cos(a)))*x - 6*x^2 + (x^3*(cos(
a) - I*sin(a)) - ((cos(a) - I*sin(a))*cos(2*a) + (I*cos(a) + sin(a))*sin(2*a))*x)*log(x^2 + 2*x*cos(a) + cos(a
)^2 + sin(a)^2) - (x^3*(cos(a) - I*sin(a)) - ((cos(a) - I*sin(a))*cos(2*a) - (-I*cos(a) - sin(a))*sin(2*a))*x)
*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2) + 2*cos(2*a) + 2*I*sin(2*a))/(2*x^3 - x*(2*cos(2*a) + 2*I*sin(2*a
)))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}}, x\right ) - 2}{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="fricas")

[Out]

((x*e^(2*I*a - 2*log(x)) - x)*integral(-(e^(2*I*a - 2*log(x)) + 1)/(x^2*e^(2*I*a - 2*log(x)) - x^2), x) - 2)/(
x*e^(2*I*a - 2*log(x)) - x)

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Sympy [A]  time = 1.00885, size = 44, normalized size = 0.69 \begin{align*} \frac{3 x^{2} - e^{2 i a}}{x^{3} - x e^{2 i a}} - \left (- \log{\left (x - e^{i a} \right )} + \log{\left (x + e^{i a} \right )}\right ) e^{- i a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*ln(x))**2/x**2,x)

[Out]

(3*x**2 - exp(2*I*a))/(x**3 - x*exp(2*I*a)) - (-log(x - exp(I*a)) + log(x + exp(I*a)))*exp(-I*a)

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Giac [A]  time = 1.25604, size = 117, normalized size = 1.83 \begin{align*} 2 \,{\left (\frac{\arctan \left (\frac{x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right ) e^{\left (-2 i \, a\right )}}{\sqrt{-e^{\left (2 i \, a\right )}}} + \frac{x e^{\left (-2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac{5 \, x^{2}}{x^{3} - x e^{\left (2 i \, a\right )}} - \frac{e^{\left (2 i \, a\right )}}{x^{3} - x e^{\left (2 i \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="giac")

[Out]

2*(arctan(x/sqrt(-e^(2*I*a)))*e^(-2*I*a)/sqrt(-e^(2*I*a)) + x*e^(-2*I*a)/(x^2 - e^(2*I*a)))*e^(2*I*a) + 5*x^2/
(x^3 - x*e^(2*I*a)) - e^(2*I*a)/(x^3 - x*e^(2*I*a))